The Arrow of Time

 

I put the water in a mug and place the mug in the microwave. It takes 2 minutes on high to bring it to a proper boil. The microwave beeps and I drop the tea bag in, Earl Grey of course, listening for the sound of bubbling as the tea bag creates additional nucleation sites within the water.

The tea piping hot, I pull it out and set it on the counter to steep.

Suppose I play that back in reverse now.

I take the piping hot tea from the counter. I place it in the microwave and pull out the tea bag. The microwave beeps and the clock runs up to 2 minutes while the tea inside cools from boiling to room temperature. The microwave radiation inside removes the heat and stores it as electricity. The water now room temperature I take out the cup and the water leaps up into the tap.

Why is it that it makes sense in one direction but not the other?

It all has to do with cause and effect.

The microwave is the cause of the water heating. The water heating is the effect. If you reverse them, it makes no sense.

The reason why things make sense in a particular order from cause to effect has to do with the 2nd law of thermodynamics which says that entropy, the level of disorder in a system, must always increase. Thus, a cause, it seems, must have lower entropy than its effect.

Suppose I were to drop the mug of tea on the floor. The mug smashes into pieces. The tea sprays out over the floor. The entropy of the mug and the tea has increased by being allowed to smash and spill. One must be the cause of the other.

Thermodynamics was developed in the late 18th century and boils down into a set of laws. The two main laws are these:

1. Energy cannot be created or destroyed.
2. Entropy must always increase with time.

Law #1 is an absolute law of physics. It is strictly obeyed, even in quantum mechanics.

Law #2 however is a “de facto” law of physics. It is not strictly obeyed, but only obeyed on average. Now, averages in thermodynamics tend to be pretty solid because, if every particle is making bets and there are trillions of trillions of particles, then the law of large numbers says that the average is as good as law. But, it is still not a strict law of physics.

Instead, the laws of physics that are strict are all “time reversible” meaning that causes and effects are actually interchangeable. Theoretically, in a closed system, it is possible for the cup to leap up and reassemble itself. It is just not probable.

For any physics enthusiast, this is all common knowledge, but what isn’t commonly known is why it must be so. Why does entropy increase in one direction and not the other? If all physical processes are time reversible, then it should allow some processes to increase in entropy in one direction while others increase in entropy in the reverse time direction. Yet, we do not see this actually happen. All physical processes increase in entropy in only one direction.

The overarching question is: why?

This problem is called the Problem of the Arrow of Time.

Quite a few solutions have been proposed to the dilemma:

1. The universe simply began in a low entropy state. This is ad hoc and hardly explains why entropy increase is always observed.
2. The expansion of the universe forces the thermodynamic arrow of time in one direction. Why this should be so is unknown, since the thermodynamic arrow applies to all systems no matter how small.
3. Physical laws are incomplete and should reflect irreversible processes, things that are not time symmetric. Current physical laws depend on this symmetry, so it would be a major breakthrough to find that time symmetry is not respected (this would be called a Charge-Parity-Time or CPT violation and would be as impactful as showing that Lorentz symmetry is violated).
4. Quantum decoherence — this interaction of pure quantum states (like isolated particles) with macroscopic objects — causes the arrow of time. To me this is just moving the question from entropy to decoherence. Why decohere in one direction and not the other when quantum mechanics is time reversible?

The most powerful resolution to the problem, however, focuses not on a defect of physical law but on us, or more properly, any information-based system.

This is the information or memory-based arrow of time resolution which states that the information content of a system, i.e., its memory increases with increasing entropy. Thus, all information storage systems experience time as flowing towards increasing entropy despite time being reversible.

Like the Silence from Doctor Who, any processes that evolve towards lower entropy are not remembered.

In 1950 Erwin Schroedinger, one of the founders of quantum mechanics, described this point of view using statistical mechanics. Suppose you have a system in non-equilibrium, such as a crystal that has gained energy and is in the process of melting into a liquid. Now, before it has reached an equilibrium, you divide it into four isolated pieces.

Schroedinger showed that each of these systems will pick a time direction and evolve to equilibrium in that direction. The evolution of the closed and isolated system will define a direction of time for itself. Therefore, the time t of Einstein, Newton, and even Boltzmann is different from the phenomenological, that is, observed, arrow of time which is t or -t. Unlike physical time, phenomenological time, the time we experience, is an entropy gradient, a path of increase that is defined through spacetime.

The information theory of time combines the entropy gradient arrow of Schroedinger with the information theory of Shannon.

Claude Shannon was just another researcher at Bell Labs working on the telephone system. Given that the phone system is intimately related to the transmission of information, he started thinking about how to characterize information in terms of entropy. He developed a theory that entropy is related to the number of bits, binary 0’s and 1’s, that a system can carry. This has to do with the number of different configurations a system can have. A low entropy system can only have a few configurations, so only a few bits. A high entropy system can have many configurations, so many bits.

This relationship suggests that the flow of time we perceive and the entropy gradient are related to the direction information grows. More importantly, it also suggests that while entropy growth adds memories, entropy reduction takes them away.

This theory was made more secure using quantum information theory in 2009 in a stunning paper by Italian researcher Lorenzo Maccone. Maccone showed that phenomenological time can proceed in either time direction as Schroedinger indicated and as time reversibility dictates. Quantum mechanics, however, says that processes that decrease entropy cannot leave any trace that they ever happened. That is, entropy decreasing processes actually remove the fact that they happened as if time were running in reverse.

Maccone works out his theory using the mathematics of quantum mechanics, using wavefunctions in Hilbert spaces (infinite dimensional spaces that guarantee a 2-way Fourier transform exists). To illustrate the idea, Maccone offers the following thought experiment:

Alice is in a lab that is perfectly isolated so that she evolves as a complete quantum state.

Bob sends her some energy in the form of light.

Alice detects the energy with detectors that heat up when exposed to the light. She is not making any measurements of Bob’s light modes because the energy has been converted into entropy, as thermodynamic heat.

Bob wants his energy back, i.e., wants to time reverse the energy process. In order to do so, he must remove the entropy. To do that, he has to remove any correlation between the energy he sent her and her instruments, including any lab notes she made. If he had that power, he could reduce her lab back to a lower entropy state and recover the energy. But, the result would be that Alice would have no memory of the event, lab notes empty, detectors cool.

Thus, in the process of reversing entropy, a system must remove any correlations with its having happened. No correlations, no information, no memory.

In Schroedinger’s example, therefore, those of the four closed systems that run in reverse time cannot be studied by physics because it is as if their evolution did not happen at all. So rather than seeing the systems as if they were running in reverse, we would not be aware of them. Our memory and the states of any other apparatus we used to study them would disappear.

Indeed, closed systems can run in reverse time all the “time” in our universe but we have no way of detecting them because they abscond with their information content. It would be like trying to see the future.

An additional, and more strange conclusion, is that we would be able to “remember” those processes in the past and lose those memories in the future. That is, a process reversed from our own phenomenological direction removes information content (quantum correlations) about its existence, which suggests that in the past that information existed and decreased.

Could this mean that we can remember the future?

Well, yes and no, as macroscopic beings we are highly correlated with one another and one of the requirements for systems to be moving in opposite phenomenological directions is that they have to be almost perfectly isolated from one another.

Ignoring that fact, let’s go back to the cup of tea:

While the tea cup can leap up from the floor and reassemble itself in my hand, quantum mechanics does not allow me to perceive it because my brain is running in the other phenomenological direction. Instead, I would perceive the smashed cup and remember its having fallen. Then I would gradually lose my memory of its falling as it leapt into the air until I’m holding the cup with no memory that it fell at all.

If you’ve seen the movie Memento, you’ll know what I’m talking about.

Maccone concludes that the universe itself may be in a zero-entropy state but we perceive its evolution from low to high entropy because of the trick of information systems.

There may, in fact, be another universe overlapping ours but isolated evolving backwards in time from us that we cannot perceive. Perhaps there are entire alien worlds out there, beyond the observable universe, evolving backwards.

 Source:

Brown, Harvey R., and Jos Uffink. “The origins of time-asymmetry in thermodynamics: The minus first law.” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 32.4 (2001): 525–538.

Maccone, Lorenzo. “Quantum solution to the arrow-of-time dilemma.” Physical review letters 103.8 (2009): 080401.